# COMPARING AND ORDERING IRRATIONAL NUMBERS

How we are comparing and ordering rational and irrational numbers ?

The given numbers may in the form of fraction, percentage, decimals, square roots  etc. We convert the given numbers into decimal and compare them.

Let us understand what are rational and irrational numbers.

What is rational number ?

All numbers that can be written in the form of p/q is rational number.

What are irrational numbers ?

The numbers that cannot be written in the form of fraction, those are irrational numbers.

## Example Problems on Comparing and Ordering Irrational Numbers

List the following numbers in order from least to greatest.

Problem 1 :

39/8, 4.2, √16

Solution :

39/8, 4.2, √16

4.875, 4.2, 4

So, the numbers in order from least to greatest is

√16 < 4.2 < 39/8 .

Problem 2 :

√24, √33, 5.1

Solution :

√24, √33, 5.1

Comparing √24 and √33, √33 is greater.

√24 is lesser than √25, so approximate value of √24 is 4....

√24, 5.1, √33

So, the numbers in order from least to greatest is

√24 < 5.1 <√33.

Problem 3 :

√100, √110, 32/7

Solution :

√100, √110, 32/7

10, 10.4, 4.57

So, the numbers in order from least to greatest is

32/7 < √100 < √110.

Problem 4 :

9.4, 19/2, √80

Solution :

9.4, 19/2, √80

9.4, 9.5, 8.944

So, the numbers in order from least to greatest is

√80 < 9.4 < 19/2.

Problem 5 :

√35, √32, √37, 22/3

Solution :

Comparing radicals √35, √32 and √37

√32 is least

√35 is greater than √32

√37 is greater than √35

22/3 is equal 7.33

So, the numbers in order from least to greatest is

√32 <√35 <√37< 22/3.

Problem 6 :

3.5, √10, 13/3,√15

Solution :

3.5, √10, 13/3, √15

Comparing √10 and √15

√15 > √10

√9 < √10 < √15 < √16

13/3 = 4.....

So, approximate value of √10 and √15 lies between 3 and 4. √10 is nearer to √9, then the value of √10 does not exceed 3.5

So, the numbers in order from least to greatest is

√10 < 3.5 <√15 <13/3.

Problem 7 :

√65, √60, 8.5, 37/4

Solution :

√65, √60, 8.5, 37/4

Comparing √60 and √65, √65 is greater.

√60 = 7......

√65 = 8.....

37/4 = 9.25

So, the numbers in order from least to greatest is

√60 < √65 < 8.5 < 37/4.

Problem 8 :

√39, √25, 5.3, √26, 23/4

Solution :

√39, √25, 5.3, √26, 23/4

Comparing √25, √26 and √39

√25 < √26 < √39

Considering √39, it is greater than √36. So it is more than 6.

23/4 = 5.75

7.6672, 5, 5.3, 5.099, 5.75

So, the numbers in order from least to greatest is

√25 < √26 < 5.3 < 23/4 < √39.

Problem 9 :

√12, √15, 4.3, √9, 14/5

Solution :

√12, √15, 4.3, √9, 14/5

Comparing √15, √12 and √9

√9 < √12 < √15

Here √15 is nearer to √16. So, it's value must be 3.8.....

14/5 = 2.8

So, the numbers in order from least to greatest is

14/5 < √9 < √12 < √15 < 4.3.

Problem 10 :

√49, √63, 7.3, √38, 15/2

Solution :

√49, √63, 7.3, √38, 15/2

Comparing √49, √38 and √63

√38 < √49 < √63

Comparing √63, √63 is nearer to √64. So, its value will be 7.8 or 7.9

15/2 = 7.5

So, the numbers in order from least to greatest is

√38 < √49 < 7.3, 15/2 < √63.

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